Quantitative surgery and total mean curvature

TL;DR

The paper develops a quantitative surgery framework that extends classical Gromov–Lawson and Lawson–Michelsohn techniques to control mean curvature during cobordisms. It proves a Gromov-type bound on the total mean curvature of fill-ins with a fixed scalar curvature lower bound, under a spin assumption on the boundary manifold, by reducing the problem to the sphere via high-codimension surgeries and applying Shi–Wang–Wei’s total mean curvature bound. The authors establish stronger results, including a weak fixed-H bound, a uniform bound for metric families, and a construction of a new quasi-local mass m_σ(M) that is nonnegative for spin M and recovers Brown–York mass in special low-dimensional cases. These results unify surgery-based control of scalar and mean curvatures with quasi-local mass concepts, offering robust tools for understanding geometric invariants of fill-ins and potential applications in general relativity.

Abstract

We develop quantitative surgery, which extends the classical constructions of Gromov--Lawson and Lawson--Michelsohn. As an application, we prove a conjecture of Gromov on the total mean curvature of fill-ins.

Figures (9)

• Figure 1: A cross-section of a family of Gromov--Lawson handles $\Sigma_\varrho$ attached to $\Omega$ along $M=\partial \Omega$. The origin corresponds to the intersection of the cross-section with $S$.
• Figure 2: The disjoint embeddings $\psi_1$ and $\psi_2$ which are isotopic to $\psi$ as small displacements.
• Figure 3: The cobordism $W$ consisting of one handle and the embedded handle
• Figure 4: Deformation of $g_\Omega$ near $\partial \Omega$ (grey region)
• Figure 5: The local deformation $\psi$ for $t=1$.

Theorems & Definitions (29)

• Conjecture 1.1: Gromov2023FourLectures
• Theorem 1.2
• Remark 1.3
• Theorem 2.1
• Definition 3.1
• Proposition 3.2
• proof
• Remark 4.2
• Proposition 4.3
• proof